Behavior of Fréchet mean and Central Limit Theorems on spheres

Do Tran

Published 2019-11-05Version 1

Jacobi fields are used to compute higher derivatives of the Fr\'{e}chet function on spheres with absolutely continuous rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fr\'{e}chet mean; (ii) a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.